MathDB
Including chain of minimal sets under some valuation

Source: China MO 2024, Day 1, Problem 3

November 28, 2023
combinatorics

Problem Statement

Let p⩾5p \geqslant 5 be a prime and S={1,2,…,p}S = \left\{ 1, 2, \ldots, p \right\}. Define r(x,y)r(x,y) as follows: r(x,y)={y−xy⩾xy−x+py<x. r(x,y) = \begin{cases} y - x & y \geqslant x \\ y - x + p & y < x \end{cases}. For a nonempty proper subset AA of SS, let f(A)=∑x∈A∑y∈A(r(x,y))2.f(A) = \sum_{x \in A} \sum_{y \in A} \left( r(x,y) \right)^2.A good subset of SS is a nonempty proper subset AA satisfying that for all subsets B⊆SB \subseteq S of the same size as AA, f(B)⩾f(A)f(B) \geqslant f(A). Find the largest integer LL such that there exists distinct good subsets A1⊆A2⊆…⊆ALA_1 \subseteq A_2 \subseteq \ldots \subseteq A_L.
Proposed by Bin Wang
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